Related papers: Irreducibility and singularities of some nested Hi…
We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is an unramified regular local ring, then just as in the equicharacteristic case, the intersection…
Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group $\pi_1(S^3 \setminus \operatorname{WD}^n(K))$ is bounded below by $n+1$. As an application, we show that…
Let F(X,n):= X^n-\Delta be the complementary of the union \Delta of the diagonals of X^n and let U be a quotient of F(X,n) (possibly trivial) by a subgroup of the symmetric group S_n. We construct compactifications of U in products of…
We show that the projection morphism $X^{[3,4]} \lra X^{[3]}$ is flat even if it has reducible fiber. After analyzing blow-up constructions related to $X^{[3,4]}$, we conclude that $X^{[3,4]}$ has canonical Gorenstein singularities. As a…
We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds…
In math.AG/0108089 we gave sufficient conditions for the irreducibility of the family V of irreducible curves in the linear system |D| with precisely r singular points of topological respectively analytical types S1,...,Sr on several…
For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the disconnected $k$-vertex induced subgraphs of $G$. Bayer, Denker, Jeli\'c Milutinovi\'c, Sundaram, and Xue proved that the…
We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with…
We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical…
We determine the automorphism group of the Hilbert scheme of two points on a generic projective K3 surface of any polarization. We obtain in particular new examples of Hilbert schemes of points having non-natural non-symplectic…
We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the…
Given an algebraically closed field $K$ of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of $\mathbb P^2(K)$. Answering…
The space of monic centered cubic polynomials with marked critical points is isomorphic to C^2. For each n>0, the locus Sn formed by all polynomials with a specified critical point periodic of exact period n forms an affine algebraic set.…
Given a smooth curve on a smooth surface, the Hilbert scheme of the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural…
For infinitely many $d$, Hassett showed that special cubic fourfolds of discriminant $d$ are related to polarized K3 surfaces of degree $d$ via their Hodge structures. For half of the $d$, each associated K3 surface $(S,L)$ canonically…
We analyse the geometry of Hilbert schemes of points on abelian surfaces and Beauville's generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of…
In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for…
The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such…
The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand…
In this paper we show that any irreducible finite dimensional representation of $SL_{n+1}$ remains indecomposable if restricted to n--dimensional abelian subalgebras spanned by simple root vectors.